Peel the potatoes and the carrot, cut them into large chunks and cook them until half done.
Meanwhile, trim and wash the fish. Dry it with paper towels and make diagonal cuts in each side.
Turn on the oven to medium-high heat.
In a food processor chop garlic, parsley, dill, butter, lemon juice and zest, until you get a smooth mixture.
Rub the green butter mixture into the fish, make sure to rub some of it into the belly and the cuts you made.Put the fish in a large baking dish and season it with some salt and pepper.
When the potatoes and the carrot are half done, drain them and mix them with some of the green butter mixture. Add them to the dish with the fish. Season the vegetables with some more salt and pepper, pour in the coconut milk and put the dish in the oven.
Bake in the oven for 40 min on medium high heat. In the mid time carefully turn the fish on the other side and stir the vegetables, so they won't be too dry. Serve hot!